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Simplifying 100 = X(X + 1) Reorder the terms: 100 = X(1 + X) 100 = (1 * X + X * X) 100 = (1X + X2) Solving 100 = 1X + X2 Solving for variable 'X'. Reorder the terms: 100 + -1X + -1X2 = 1X + -1X + X2 + -1X2 Combine like terms: 1X + -1X = 0 100 + -1X + -1X2 = 0 + X2 + -1X2 100 + -1X + -1X2 = X2 + -1X2 Combine like terms: X2 + -1X2 = 0 100 + -1X + -1X2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -100 + X + X2 = 0 Move the constant term to the right: Add '100' to each side of the equation. -100 + X + 100 + X2 = 0 + 100 Reorder the terms: -100 + 100 + X + X2 = 0 + 100 Combine like terms: -100 + 100 = 0 0 + X + X2 = 0 + 100 X + X2 = 0 + 100 Combine like terms: 0 + 100 = 100 X + X2 = 100 The X term is X. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + X2 = 100 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + X2 = 100 + 0.25 Combine like terms: 100 + 0.25 = 100.25 1.25 + X2 = 100.25 Factor a perfect square on the left side: (X + 0.5)(X + 0.5) = 100.25 Calculate the square root of the right side: 10.012492197 Break this problem into two subproblems by setting (X + 0.5) equal to 10.012492197 and -10.012492197.Subproblem 1
X + 0.5 = 10.012492197 Simplifying X + 0.5 = 10.012492197 Reorder the terms: 0.5 + X = 10.012492197 Solving 0.5 + X = 10.012492197 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + X = 10.012492197 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + X = 10.012492197 + -0.5 X = 10.012492197 + -0.5 Combine like terms: 10.012492197 + -0.5 = 9.512492197 X = 9.512492197 Simplifying X = 9.512492197Subproblem 2
X + 0.5 = -10.012492197 Simplifying X + 0.5 = -10.012492197 Reorder the terms: 0.5 + X = -10.012492197 Solving 0.5 + X = -10.012492197 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + X = -10.012492197 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + X = -10.012492197 + -0.5 X = -10.012492197 + -0.5 Combine like terms: -10.012492197 + -0.5 = -10.512492197 X = -10.512492197 Simplifying X = -10.512492197Solution
The solution to the problem is based on the solutions from the subproblems. X = {9.512492197, -10.512492197}
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